Triangle Congruence Theorem

Triangle congruence theorem consists of five theorems that prove the congruence of two triangles. Two triangles are said to be congruent or the same if the shape and size of both the triangles are the same i.e. the corresponding sides placed in the same position and the corresponding angles placed in the same position of both triangles are the same. Let us learn more about the triangle congruence theorem, the different theorems, and solve a few examples.

Triangle congruence theorem or triangle congruence criteria help in proving if a triangle is congruent or not. The word congruent means exactly equal in shape and size no matter if we turn it, flip it or rotate it. In geometry, if the shapes are superimposed on each other, they are termed congruent figures, for example, triangles and quadrilaterals can be congruent.

The following are the triangle congruence theorems or the triangle congruence criteria that help to prove the congruence of triangles.

• SSS (Side, Side, Side)
• SAS (Side, Angle, Side)
• ASA (Angle, Side, Angle)
• AAS (Angle, Angle, Side)
• RHS (Right angle-Hypotenuse-Side or the Hypotenuse Leg theorem)

Angle-Side-Angle (ASA) theorem states that if two triangles are considered congruent, then the two corresponding angles equal to each other will include a corresponding side that is equal to each other. Let us see the proof of the ASA theorem:

Consider the following two triangles, Δ ABC and Δ DEF.

We are given that,

BC = EF
∠B = ∠E
∠C = ∠F

Does this mean ΔABC and ΔDEF are congruent? Let us superimpose both the triangles and align them according to the angles and sides. Align EF exactly with BC. Since ∠B = ∠E, the direction of ED will be the same as the direction of BA. Similarly, since∠C = ∠F, the direction of FD will be the same as the direction of CA. This means that the point of intersection of ED and FD (which is D) will coincide exactly with the point of intersection of BA and CA (which is A). Hence, we notice that both the triangles are of the same shape and size, we can say that ΔABC ≅ ΔDEF.

The Side-Angle-Side (SAS) congruence theorem states that, if two corresponding sides of two triangles are equal including the corresponding angles formed by these sides then the two triangles are congruent. Let us see the proof of the theorem:

Given: AB=PQ, BC=QR, and ∠B=∠Q. To prove: ΔABC ≅ ΔPQR

Just like in ASA, let us superimpose triangles here again. Place the triangle ΔABC over the triangle ΔPQR such that B falls on Q and side AB falls along the side PQ. Since AB=PQ, so point A falls on point P, ∠B=∠Q, so the side BC will fall along the side QR, and BC=QR, so point C falls on point R. Thus, BC coincides with QR and AC coincides with PR. Therefore, we can say ΔABC ≅ ΔPQR.

Side-Side-Side (SSS) congruence theorem states that if three sides of a triangle is equal to the corresponding sides of the other triangle, the two triangles are said to be congruent. Let us see the proof of the theorem:

Given: AB = DE, BC = EF, and AC = DF. To prove: ∆ABC ≅ ∆DEF.

We know that the three sides of both the triangles are of the same size and length. When we superimpose both the triangles, DE will be placed on AB, EF will be placed on BC, and DF will be placed on AC. Which makes it AB = DE, BC = EF, and AC = DF. Therefore, we can say that ∆ABC ≅ ∆DEF.

Angle-Angle-Side congruence theorem states that if two angles of a triangle with a side that is not included between the angles is equal to the corresponding angles and side of the other triangle, they are considered to be congruent. Let us see the proof of the theorem:

Given: AB = DE, ∠B=∠E, and ∠C=∠F. To prove: ∆ABC ≅ ∆DEF

If both the triangles are superimposed on each other, we see that ∠B=∠E and ∠C=∠F. And the non-included sides AB and DE are equal in length. Therefore, we can say that ∆ABC ≅ ∆DEF.

RHS, Right angle-Hypotenuse-Side or the Hypotenuse Leg theorem, states that if the hypotenuse and side of one right-angled triangle are equal to the hypotenuse and the corresponding side of another right-angled triangle, the two triangles are congruent. When we keep the hypotenuse and any one of the other 2 sides of two right triangles equal, we are automatically getting three similar sides, as all three sides in a right triangle are related to each other and that relation is popularly known as Pythagoras theorem. Let us see the proof of the theorem:

Given: Hypotenuse side and angles are equal. Tp prove: ∆ABC ≅ ∆PQR

If in any two triangles, the hypotenuse side doesn't measure the same or doesn't have a measurement we use (hypotenuse)² = (base)² + (perpendicular)². Both the triangles need to have a 90° right angle equal and the hypotenuse to be equal. In the above image, as we can see the measurements are the same for both triangles. Hence, we can say ∆ABC ≅ ∆PQR.

Related Topics

Listed below are a few topics related to the triangle congruence theorem, take a look:

• CPCTC
• Side-Angle-Side Formula
• Same Side Interior Angles

What is Meant by Triangle Congruence Theorem?

Triangle congruence theorem or triangle congruence criteria help in proving if a triangle is congruent or not. There are 5 triangle congruence theorems - Side Side Side Theorem, Side Angle Side Theorem, Angle Side Angle Theorem, Angle Angle Side Theorem, and Right angle-Hypotenuse-Side or the Hypotenuse Leg theorem.

What is the 4 Triangle Congruence Theorem?

The 4 triangle congruence theorems that help in finding if the triangles are congruent or not are:

• SSS (Side, Side, Side)
• SAS (Side, Angle, Side)
• ASA (Angle, Side, Angle)
• AAS (Angle, Angle, Side)

What is SSS, SAS, ASA, and AAS Triangle Congruence Theorem?

The 4 different triangle congruence theorems are:

• SSS: Where three sides of two triangles are equal to each other.
• SAS: Where two sides and an angle included in between the sides of two triangles are equal to each other.
• ASA: Where two angles along with a side included in between the angles of any two triangles are equal to each other.
• AAS: Where two angles of any two triangles along with a side that is not included in between the angles, are equal to each other.

What is the RHS Triangle Congruence Theorem?

RHS, Right angle-Hypotenuse-Side or the Hypotenuse Leg theorem, states that if the hypotenuse and side of one right-angled triangle are equal to the hypotenuse and the corresponding side of another right-angled triangle, the two triangles are congruent.

How Do You Tell if a Triangle is ASA or AAS?

Both the triangle congruence theorems deal with angles and sides but the difference between the two is ASA deals with two angles with a side included in between the angles of any two triangles. Whereas AAS deals with two angles with a side that is not included in between the two angles of any two given triangles.

What are the Properties of Triangle Congruence Theorem?

There are 5 properties of congruent triangles which form the conditions to determine if they are congruent or not. They are:

• SSS Criterion for Congruence
• SAS Criterion for Congruence
• ASA Criterion for Congruence
• AAS Criterion for Congruence
• RHS Criterion for Congruence